The Moduli Problem for Plane Branches
About this Title
Oscar Zariski. Translated by Ben Lichtin
Publication: University Lecture Series
Publication Year 2006: Volume 39
ISBNs: 978-0-8218-2983-7 (print); 978-1-4704-2183-0 (online)
MathSciNet review: MR2273111
MSC: Primary 14H20; Secondary 01A75, 14H45
Moduli problems in algebraic geometry date back to Riemann's famous count of the $3g-3$ parameters needed to determine a curve of genus $g$. In this book, Zariski studies the moduli space of curves of the same equisingularity class. After setting up and reviewing the basic material, Zariski devotes one chapter to the topology of the moduli space, including an explicit determination of the rare cases when the space is compact. Chapter V looks at specific examples where the dimension of the generic component can be determined through rather concrete methods. Zariski's last chapter concerns the application of deformation theory to the moduli problem, including the determination of the dimension of the generic component for a particular family of curves.
An appendix by Bernard Teissier reconsiders the moduli problem from the point of view of deformation theory. He gives new proofs of some of Zariski's results, as well as a natural construction of a compactification of the moduli space.
Graduate students and research mathematicians interested in algebraic geometry, especially moduli questions, and singularities.
Table of Contents
- Chapter 1. Preliminaries
- Chapter 2. Equisingularity invariants
- Chapter 3. Parametrizations
- Chapter 4. The moduli space
- Chapter 5. Examples
- Chapter 6. Applications of deformation theory
- Appendix by B. Teissier
- Chapter I. The monomial curve $C^\Gamma $ and its formations
- Chapter II. Application to the study of the moduli space of a branch